A conservation–prioritized approach simultaneously enhancing mass and momentum conservation of least-squares method for Stokes/Navier–Stokes problems

Abstract

In solving the system of Stokes and Navier–Stokes equations, lack of mass conservation has been viewed as the critical drawback of the finite element methods based on the least-squares (LS) principle. Although many modifications have been proposed, there is a need for a global approach that improves both mass conservation and momentum conservation. The key to such a global method is to control local conservation, which is weaker than to control the residual everywhere. Accordingly, a new method named conservation–prioritized​ Moment Least-Squares (CMLS) is developed. The CMLS method emerges from the moment conditions. Among these moment conditions, the zero-order moment, which exactly expresses the local conservation condition on the element, is prioritized over the others; thereby, good local conservation can be achieved. The advantages of the CMLS method over the LS method are demonstrated by conservation errors, convergence studies, and numerical accuracy in nonlinear Navier–Stokes tests. Besides, the CMLS method retains the merits of the LS method: it has a symmetric positive-definite global matrix and the same interpolation for both velocity and pressure.